The Hankel transformation on $M’_ \mu$ and its representation

Abstract:

The Hankel transformation was extended by Zemanian to certain generalized functions of slow growth through a generalization of Parseval’s equation as \begin{equation}\tag {$(1)$} \langle {h_\mu }f,\varphi \rangle = \langle f,{h_\mu }\varphi \rangle \end{equation} where $\varphi , {h_\mu }\varphi \in {H_\mu }, f \in {H’_\mu }$. Later, Koh and Zemanian defined the generalized complex Hankel transformation on ${J_\mu } = {\bigcup }_{\nu = 1}^\infty {J_{{a_\nu },\mu }}$, where ${J_{{a_\nu },\mu }}$ is the testing function space which contains the kernel function, $\sqrt {xy} {J_\mu }(xy)$. A transformation was defined directly as the application of a generalized function to the kernel function, i.e., for $f \in {J’_\mu }$, \begin{equation}\tag {$(2)$} ({h_\mu }f)(y) = \langle f(x),\sqrt {xy} {J_\mu }(xy)\rangle .\end{equation} In this paper, we extend definition (2) to a larger space of generalized functions. We first introduce the test function space ${M_{a,\mu }}$ which contains the kernel function and show that ${H_\mu } \subset {M_{a,\mu }} \subset {J_{a,\mu }}$. We then form the countable union space ${M_\mu } = {\bigcup }_{\nu = 1}^\infty {M_{{a_\nu },\mu }}$ whose dual ${M’_\mu }$ has ${J’_\mu }$ as a subspace. Our main result is an inversion theorem stated as follows. Let $F(y) = ({h_\mu }f)(y) = \langle f(x),\sqrt {xy} {J_\mu }(xy)\rangle ,f \in {M’_\mu }$, where y is restricted to the positive real axis. Let $\mu \geq - \frac {1}{2}$. Then, in the sense of convergence in ${H’_\mu }$, \[ f(x) = \lim \limits _{r \to \infty } \int _0^r {F(y)} \sqrt {xy} {J_\mu }(xy)dy.\] This convergence gives a stronger result than the one obtained by Koh and Zemanian (1968). Secondly, we prove that every generalized function belonging to ${M’_{a,\mu }}$ can be represented by a finite sum of derivatives of measurable functions. This proof is analogous to the method employed in structure theorems for Schwartz distributions (Edwards, 1965), and similar to one by Koh (1970).